The vector(s) which is/are coplanar with vectors \hat{i} + \hat{j} + 2 \hat{k} \textrm{ }\textrm{ } and \textrm{ }\textrm{ } \hat{i} + 2 \hat{j} + \hat{k} and perpendicular to the vector \hat{i} + \hat{j} + \hat{k} is/are

Engineering

Mathematics

Vector Triple Product

Question

The vector(s) which is/are coplanar with vectors $\hat{i} + \hat{j} + 2 \hat{k} and \hat{i} + 2 \hat{j} + \hat{k}$ and perpendicular to the vector $\hat{i} + \hat{j} + \hat{k}$ is/are

$\hat{j} - \hat{k}$

$\hat{i} - \hat{j}$

$- \hat{i} + \hat{j}$

$- \hat{j} + \hat{k}$

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$\bar{r} = x \hat{i} + y \hat{j} + z \hat{k}$ is coplanar with the given vector so

$∴ | \begin{matrix} x & y & z \\ 1 & 1 & 2 \\ 1 & 2 & 1 \end{matrix} | = 0$

So, 3x = y + z .....(1)

$∴ \bar{r} ⊥ \hat{i} + \hat{j} + \hat{k}$

So, $\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 0$

So, x + y + z = 0 .....(2)

On solving (1) & (2)

So, x = 0 $∴$ y + z = 0 $∴$ (A) & (D) Satisfy